
Function Spaces, 1
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This is the first part of the second revised and extended edition of the well established book "Function Spaces" by Alois Kufner, Oldrich John, and Svatopluk Fucík. Like the first edition this monograph is an introduction to function spaces defined in terms of differentiability and integrability classes. It provides a catalogue of various spaces and benefits as a handbook for those who use function spaces in their research or lecture courses.
This first volume is devoted to the study of function spaces, based on intrinsic properties of a function such as its size, continuity, smoothness, various forms of a control over the mean oscillation, and so on. The second volume will be dedicated to the study of function spaces of Sobolev type, in which the key notion is the weak derivative of a function of several variables.
Reviews / Votes
"Das Buch stellt dadurch eine nützliche Informationsquelle (ergänzt durch ein umfangreiches Literaturverzeichnis)dar, in einem nicht besonders übersichtlichen Teilgebiet."
V. Losert in: Monatsh Math 186 (2018), 561-564
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Content
2 - 1 Preliminaries [Seite 17]
2.1 - 1.1 Vector space [Seite 17]
2.2 - 1.2 Topological spaces [Seite 18]
2.3 - 1.3 Metric, metric space [Seite 22]
2.4 - 1.4 Norm, normed linear space [Seite 22]
2.5 - 1.5 Modular spaces [Seite 23]
2.6 - 1.6 Inner product, inner product space [Seite 26]
2.7 - 1.7 Convergence, Cauchy sequences [Seite 27]
2.8 - 1.8 Density, separability [Seite 28]
2.9 - 1.9 Completeness [Seite 28]
2.10 - 1.10 Subspaces [Seite 29]
2.11 - 1.11 Products of spaces [Seite 30]
2.12 - 1.12 Schauder bases [Seite 30]
2.13 - 1.13 Compactness [Seite 31]
2.14 - 1.14 Operators (mappings) [Seite 32]
2.15 - 1.15 Isomorphism, embeddings [Seite 34]
2.16 - 1.16 Continuous linear functionals [Seite 35]
2.17 - 1.17 Dual space, weak convergence [Seite 36]
2.18 - 1.18 The principle of uniform boundedness [Seite 37]
2.19 - 1.19 Reflexivity [Seite 37]
2.20 - 1.20 Measure spaces: general extension theory [Seite 38]
2.21 - 1.21 The Lebesgue measure and integral [Seite 45]
2.22 - 1.22 Modes of convergence [Seite 50]
2.23 - 1.23 Systems of seminorms, Hahn-Saks theorem [Seite 52]
3 - 2 Spaces of smooth functions [Seite 54]
3.1 - 2.1 Multiindices and derivatives [Seite 54]
3.2 - 2.2 Classes of continuous and smooth functions [Seite 55]
3.3 - 2.3 Completeness [Seite 59]
3.4 - 2.4 Separability, bases [Seite 61]
3.5 - 2.5 Compactness [Seite 67]
3.6 - 2.6 Continuous linear functionals [Seite 71]
3.7 - 2.7 Extension of functions [Seite 75]
4 - 3 Lebesgue spaces [Seite 78]
4.1 - 3.1 Lp-classes [Seite 78]
4.2 - 3.2 Lebesgue spaces [Seite 82]
4.3 - 3.3 Mean continuity [Seite 83]
4.4 - 3.4 Mollifiers [Seite 85]
4.5 - 3.5 Density of smooth functions [Seite 87]
4.6 - 3.6 Separability [Seite 87]
4.7 - 3.7 Completeness [Seite 88]
4.8 - 3.8 The dual space [Seite 90]
4.9 - 3.9 Reflexivity [Seite 94]
4.10 - 3.10 The space L8 [Seite 94]
4.11 - 3.11 Hardy inequalities [Seite 99]
4.12 - 3.12 Sequence spaces [Seite 108]
4.13 - 3.13 Modes of convergence [Seite 109]
4.14 - 3.14 Compact subsets [Seite 110]
4.15 - 3.15 Weak convergence [Seite 111]
4.16 - 3.16 Isomorphism of Lp(O) and Lp(0, µ(O)) [Seite 112]
4.17 - 3.17 Schauder bases [Seite 113]
4.18 - 3.18 Weak Lebesgue spaces [Seite 117]
4.19 - 3.19 Remarks [Seite 120]
5 - 4 Orlicz spaces [Seite 124]
5.1 - 4.1 Introduction [Seite 124]
5.2 - 4.2 Young function, Jensen inequality [Seite 125]
5.3 - 4.3 Complementary functions [Seite 131]
5.4 - 4.4 The .2-condition [Seite 135]
5.5 - 4.5 Comparison of Orlicz classes [Seite 138]
5.6 - 4.6 Orlicz spaces [Seite 142]
5.7 - 4.7 Hölder inequality in Orlicz spaces [Seite 147]
5.8 - 4.8 The Luxemburg norm [Seite 150]
5.9 - 4.9 Completeness of Orlicz spaces [Seite 153]
5.10 - 4.10 Convergence in Orlicz spaces [Seite 154]
5.11 - 4.11 Separability [Seite 159]
5.12 - 4.12 The space EF(O) [Seite 161]
5.13 - 4.13 Continuous linear functionals [Seite 167]
5.14 - 4.14 Compact subsets of Orlicz spaces [Seite 171]
5.15 - 4.15 Further properties of Orlicz spaces [Seite 177]
5.16 - 4.16 Isomorphism properties, Schauder bases [Seite 179]
5.17 - 4.17 Comparison of Orlicz spaces [Seite 182]
6 - 5 Morrey and Campanato spaces [Seite 189]
6.1 - 5.1 Introduction [Seite 189]
6.2 - 5.2 Marcinkiewicz spaces [Seite 189]
6.3 - 5.3 Morrey and Campanato spaces [Seite 192]
6.4 - 5.4 Completeness [Seite 194]
6.5 - 5.5 Relations to Lebesgue spaces [Seite 194]
6.6 - 5.6 Some lemmas [Seite 197]
6.7 - 5.7 Embeddings [Seite 201]
6.8 - 5.8 The John-Nirenberg space [Seite 203]
6.9 - 5.9 Another definition of the space JN(Q) [Seite 210]
6.10 - 5.10 Spaces Np [Seite 213]
6.11 - 5.11 Miscellaneous remarks [Seite 215]
7 - 6 Banach function spaces [Seite 219]
7.1 - 6.1 Banach function spaces [Seite 219]
7.2 - 6.2 Associate space [Seite 225]
7.3 - 6.3 Absolute continuity of the norm [Seite 232]
7.4 - 6.4 Reflexivity of Banach function spaces [Seite 239]
7.5 - 6.5 Separability in Banach function spaces [Seite 244]
8 - 7 Rearrangement-invariant spaces [Seite 253]
8.1 - 7.1 Nonincreasing rearrangements [Seite 253]
8.2 - 7.2 Hardy-Littlewood inequality [Seite 257]
8.3 - 7.3 Resonant measure spaces [Seite 259]
8.4 - 7.4 Maximal nonincreasing rearrangement [Seite 265]
8.5 - 7.5 Hardy lemma [Seite 267]
8.6 - 7.6 Rearrangement-invariant spaces [Seite 269]
8.7 - 7.7 Hardy-Littlewood-Pólya principle [Seite 271]
8.8 - 7.8 Luxemburg representation theorem [Seite 272]
8.9 - 7.9 Fundamental function [Seite 275]
8.10 - 7.10 Endpoint spaces [Seite 280]
8.11 - 7.11 Almost-compact embeddings [Seite 291]
8.12 - 7.12 Gould space [Seite 308]
9 - 8 Lorentz spaces [Seite 317]
9.1 - 8.1 Definition and basic properties [Seite 317]
9.2 - 8.2 Embeddings between Lorentz spaces [Seite 321]
9.3 - 8.3 The associate space [Seite 323]
9.4 - 8.4 The fundamental function [Seite 325]
9.5 - 8.5 Absolute continuity of norm [Seite 325]
9.6 - 8.6 Remarks on || · ||1 [Seite 327]
10 - 9 Generalized Lorentz-Zygmund spaces [Seite 329]
10.1 - 9.1 Measure-preserving transformations [Seite 329]
10.2 - 9.2 Basic properties [Seite 330]
10.3 - 9.3 Nontriviality [Seite 333]
10.4 - 9.4 Fundamental function [Seite 334]
10.5 - 9.5 Embeddings between Generalized Lorentz-Zygmund spaces [Seite 336]
10.6 - 9.6 The associate space [Seite 348]
10.7 - 9.7 When Generalized Lorentz-Zygmund space is Banach function space [Seite 369]
10.8 - 9.8 Generalized Lorentz-Zygmund spaces and Orlicz spaces [Seite 372]
10.9 - 9.9 Absolute continuity of norm [Seite 383]
10.10 - 9.10 Lorentz-Zygmund spaces [Seite 388]
10.11 - 9.11 Lorentz-Karamata spaces [Seite 389]
11 - 10 Classical Lorentz spaces [Seite 391]
11.1 - 10.1 Definition and basic properties [Seite 391]
11.2 - 10.2 Functional properties [Seite 396]
11.3 - 10.3 Embeddings [Seite 404]
11.3.1 - 10.3.1 Embeddings of type . . . [Seite 408]
11.3.2 - 10.3.2 Embeddings of type . . G [Seite 409]
11.3.3 - 10.3.3 Embeddings of type G . . [Seite 412]
11.3.4 - 10.3.4 Embeddings of type G . G [Seite 415]
11.3.5 - 10.3.5 The Halperin level function [Seite 417]
11.3.6 - 10.3.6 Embeddings of type Gp,8 (v) . .^ (w) [Seite 420]
11.3.7 - 10.3.7 The single-weight case G 1,8(v) . .1(.) [Seite 422]
11.4 - 10.4 Associate spaces [Seite 425]
11.5 - 10.5 Lorentz and Orlicz spaces [Seite 427]
11.6 - 10.6 Spaces measuring oscillation [Seite 428]
11.7 - 10.7 The missing case [Seite 441]
11.8 - 10.8 Embeddings [Seite 443]
11.8.1 - 10.8.1 Embeddings of type S . S [Seite 445]
11.8.2 - 10.8.2 Embeddings of type G . S and S . G [Seite 447]
11.8.3 - 10.8.3 Embeddings of type . . S and S . . [Seite 450]
12 - 11 Variable-exponent Lebesgue spaces [Seite 453]
12.1 - 11.1 Introduction [Seite 453]
12.2 - 11.2 Basic properties [Seite 454]
12.3 - 11.3 Embedding relations [Seite 461]
12.4 - 11.4 Density of smooth functions [Seite 463]
12.5 - 11.5 Reflexivity and uniform convexity [Seite 466]
12.6 - 11.6 Radon-Nikodým property [Seite 469]
12.7 - 11.7 Daugavet property [Seite 471]
13 - Bibliography [Seite 475]
14 - Index [Seite 488]
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